If tanθ = `(sinalpha - cosalpha)/(sinalpha + cosalpha)`, then show that sinα + cosα = `sqrt(2)` cosθ. [Hint: Express tanθ = `tan (alpha - pi/4) theta = alpha - pi/4`]

Given that: tanθ = `(sinalpha - cosalpha)/(sinalpha + cosalpha)` ⇒ tanθ = `(tanalpha - 1)/(tan alpha + 1)` = `(tanalpha - tan pi/4)/(1 + tan pi/4 tan alpha)` ⇒ tanθ = `tan(alpha - pi/4)` &there4; θ = `alpha - pi/4` ⇒ cosθ = `cos(alpha - pi/4)` ⇒ cosθ = `cos alpha cos pi/4 + sin alpha sin pi/4` ⇒ cosθ = `cos alpha . 1/sqrt(2) + sin alpha . 1/sqrt(2)` ⇒ `sqrt(2) cos theta` = cosα + sinα ⇒ sinα + cosα = `sqrt(2) cos theta` Hence proved.

If tanθ = sinα-cosαsinα+cosα, then show that sinα + cosα = 2 cosθ. [Hint: Express tanθ = tan(α-π4)θ=α-π4] - Mathematics

प्रश्न

If tanθ = $\frac{\sin α - \cos α}{\sin α + \cos α}$ , then show that sinα + cosα = $\sqrt{2}$ cosθ.

[Hint: Express tanθ = $\tan (α - \frac{π}{4}) θ = α - \frac{π}{4}$ ]

प्रमेय

उत्तर

Given that: tanθ = $\frac{\sin α - \cos α}{\sin α + \cos α}$

⇒ tanθ = $\frac{\tan α - 1}{\tan α + 1}$

= $\frac{\tan α - \tan \frac{π}{4}}{1 + \tan \frac{π}{4} \tan α}$

⇒ tanθ = $\tan (α - \frac{π}{4})$

∴ θ = $α - \frac{π}{4}$

⇒ cosθ = $\cos (α - \frac{π}{4})$

⇒ cosθ = $\cos α \cos \frac{π}{4} + \sin α \sin \frac{π}{4}$

⇒ cosθ = $\cos α . \frac{1}{\sqrt{2}} + \sin α . \frac{1}{\sqrt{2}}$

⇒ $\sqrt{2} \cos θ$ = cosα + sinα

⇒ sinα + cosα = $\sqrt{2} \cos θ$

Hence proved.

shaalaa.com

Trigonometric Functions of Sum and Difference of Two Angles

क्या इस प्रश्न या उत्तर में कोई त्रुटि है?

Q 14 Q 13 Q 15

अध्याय 3: Trigonometric Functions - Exercise [पृष्ठ ५३]

APPEARS IN

एनसीईआरटी एक्झांप्लर Mathematics [English] Class 11

अध्याय 3 Trigonometric Functions
Exercise | Q 14 | पृष्ठ ५३

वीडियो ट्यूटोरियलVIEW ALL [1]

view
वीडियो ट्यूटोरियल For All Subjects
Trigonometric Functions of Sum and Difference of Two Angles

video tutorial00:06:30

संबंधित प्रश्न

Prove the following:

$\cos (\frac{3 π}{4} + x) - \cos (\frac{3 π}{4} - x) = - \sqrt{2} \sin x$

Prove the following:

$\frac{\sin 5 x + \sin 3 x}{\cos 5 x + \cos 3 x} = \tan 4 x$

Prove the following:

$\frac{\sin x - \sin y}{\cos x + \cos y} = \tan \frac{x - y}{2}$

Prove the following:

cot x cot 2x – cot 2x cot 3x – cot 3x cot x = 1

If $\sin A = \frac{4}{5}$ and $\cos B = \frac{5}{13}$ , where 0 < A, $B < \frac{π}{2}$ , find the value of the following:
sin (A − B)

If $\sin A = \frac{4}{5}$ and $\cos B = \frac{5}{13}$ , where 0 < A, $B < \frac{π}{2}$ , find the value of the following:
cos (A − B)

If $\sin A = \frac{1}{2}, \cos B = \frac{12}{13}$ , where $\frac{π}{2}$ < A < π and $\frac{3 π}{2}$ < B < 2π, find tan (A − B).

Prove that

\frac{\cos 9^{\circ} + \sin 9^{\circ}}{\cos 9^{\circ} - \sin 9^{\circ}} = \tan 54^{\circ}

Prove that

\frac{\cos 8^{\circ} - \sin 8^{\circ}}{\cos 8^{\circ} + \sin 8^{\circ}} = \tan 37^{\circ}

Prove that: $\frac{\sin (A + B) + \sin (A - B)}{\cos (A + B) + \cos (A - B)} = \tan A$

Prove that:
$\tan \frac{π}{12} + \tan \frac{π}{6} + \tan \frac{π}{12} \tan \frac{π}{6} = 1$

Prove that:
tan 36° + tan 9° + tan 36° tan 9° = 1

Prove that sin² (n + 1) A − sin² nA = sin (2n + 1) A sin A.

If tan (A + B) = x and tan (A − B) = y, find the values of tan 2A and tan 2B.

If x lies in the first quadrant and $\cos x = \frac{8}{17}$ , then prove that:

\cos (\frac{π}{6} + x) + \cos (\frac{π}{4} - x) + \cos (\frac{2 π}{3} - x) = (\frac{\sqrt{3} - 1}{2} + \frac{1}{\sqrt{2}}) \frac{23}{17}

Find the maximum and minimum values of each of the following trigonometrical expression:

12 sin x − 5 cos x

Prove that $(2 \sqrt{3} + 3) \sin x + 2 \sqrt{3} \cos x$ lies between $- (2 \sqrt{3} + \sqrt{15}) and (2 \sqrt{3} + \sqrt{15})$

Write the maximum and minimum values of 3 cos x + 4 sin x + 5.

If tan (A + B) = p and tan (A − B) = q, then write the value of tan 2B.

If A + B + C = π, then $\frac{\tan A + \tan B + \tan C}{\tan A \tan B \tan C}$ is equal to

If $\cos P = \frac{1}{7} and \cos Q = \frac{13}{14}$ , where P and Q both are acute angles. Then, the value of P − Q is

If cot (α + β) = 0, sin (α + 2β) is equal to

\frac{\cos 10^{\circ} + \sin 10^{\circ}}{\cos 10^{\circ} - \sin 10^{\circ}} =

If tan (π/4 + x) + tan (π/4 − x) = a, then tan² (π/4 + x) + tan² (π/4 − x) =

The maximum value of $\sin^{2} (\frac{2 π}{3} + x) + \sin^{2} (\frac{2 π}{3} - x)$ is

Express the following as the sum or difference of sines and cosines:
2 cos 7x cos 3x

The value of sin(45° + θ) - cos(45° - θ) is ______.

If tanθ = $\frac{a}{b}$ , then bcos2θ + asin2θ is equal to ______.

State whether the statement is True or False? Also give justification.

If tanθ + tan2θ + $\sqrt{3}$ tanθ tan2θ = $\sqrt{3}$ , then θ = $\frac{n π}{3} + \frac{π}{9}$

If tanθ = sinα-cosαsinα+cosα, then show that sinα + cosα = 2 cosθ. [Hint: Express tanθ = tan(α-π4)θ=α-π4] - Mathematics

Advertisements

Advertisements

प्रश्न

उत्तर

APPEARS IN

वीडियो ट्यूटोरियलVIEW ALL [1]

संबंधित प्रश्न

Select a course

If tanθ = sinα-cosαsinα+cosα, then show that sinα + cosα = 2 cosθ. [Hint: Express tanθ = tan(α-π4)θ=α-π4] - Mathematics

Advertisements

Advertisements

प्रश्न

उत्तरShow Solution

APPEARS IN

वीडियो ट्यूटोरियलVIEW ALL [1]

संबंधित प्रश्न

Select a course

उत्तर